EQ-BLACK-SCHOLES · Quantitative Finance
Black–Scholes PDE
S**2*V_SS*sigma**2/2 + S*V_S*r - V*r + V_t = 0
Derivative form
S**2*sigma**2*Derivative(V(S, t), (S, 2))/2 + S*r*Derivative(V(S, t), S) - r*V(S, t) + Derivative(V(S, t), t) = 0
Variables
variable
S
current price of the underlying asset
- Object
- security
- Property
- AssetPrice
- Context
- risk_neutral_measure
variable
V
option value
- Object
- security
- Property
- OptionPrice
- Context
- risk_neutral_measure
variable
V_S
partial derivative of V with respect to underlying price S (delta)
- Object
- security
- Property
- DimensionlessRatio
- Context
- risk_neutral_measure
- Constraint
- option_delta
variable
V_SS
second partial derivative of V with respect to S (gamma)
- Object
- security
- Property
- OptionPrice
- Context
- risk_neutral_measure
- Constraint
- option_gamma
variable
V_t
partial derivative of V with respect to time
- Object
- security
- Property
- OptionPrice
- Context
- risk_neutral_measure
- Constraint
- time_derivative
variable
r
risk-free interest rate (continuous compounding)
- Object
- market
- Property
- RiskFreeRate
- Context
- risk_neutral_measure
variable
sigma
volatility of the underlying (annualised)
- Object
- market
- Property
- Volatility
- Context
- risk_neutral_measure
Axioms
classical constant_coefficients differential linear stochastic_derivation
Assumptions
- Geometric Brownian motion: dS = μS dt + σS dW for the underlying
- Constant (or deterministic) volatility and risk-free rate over the option lifetime
- No dividends, no transaction costs, no arbitrage
- European option (no early exercise); American options need free-boundary treatment
- Continuous trading, perfect liquidity, fractional shares allowed
Derivation
- Black & Scholes, J. Political Economy 81 (1973), 637 — hedging argument plus Ito's lemma
- Merton, Bell J. Econ. 4 (1973), 141 — reformulated in terms of self-financing replicating portfolio
- Under x = log(S), tau = T - t, the PDE becomes a constant-coefficient convection-diffusion equation; a further exponential substitution yields the classical heat equation
References
- Black & Scholes, J. Polit. Econ. 81 (1973), 637
- Hull, Options, Futures, and Other Derivatives, 11th ed., Ch. 15
- Wilmott, Paul Wilmott on Quantitative Finance, Ch. 5